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Description: In an engineering context, objective evaluation of vibroacoustic models is traditionally performed with visual
or numeric information. However, actual auditory perception cannot be transmitted thr...

In an engineering context, objective evaluation of vibroacoustic models is traditionally performed with visual
or numeric information. However, actual auditory perception cannot be transmitted through these types
of objective representations. Sound Field Reproduction (SFR) of sound ﬁelds emitted by physical objects
is often based on simplistic point source models with modiﬁed radiation properties or from recordings,
using stereophonic or binaural techniques. In the perspective of better perceptual evaluation of engineered
products, it would be useful if such methods would not be limited to certain types of sources, modeling
techniques or predeﬁned listening spots. A general SFR method using Wave Field Synthesis formalism
applied to common vibroacoustic models as found in mechanical engineering is proposed. SFR applied
to an analytical model of a harmonic or broadband excited plate is studied using three Secondary source
distributions geometries. Results of numerical simulations illustrate the viability and limits of the approach.

Sound Field Reproduction of Vibroacoustic

Models: Application to a Plate with Wave
Field Synthesis
Anthony Bolduc
1,2
, Philippe-Aubert Gauthier
1,2
, Telina Ramanana,
1,2
and Alain Berry
1,2
1
Groupe d’Acoustique de l’Universit´ e de Sherbrooke, Universit´ e de Sherbrooke, Sherbrooke, Canada
2
Centre for Interdisciplinary Research in Music, Media, and Technology, McGill University, Montr´ eal, Canada
Correspondence should be addressed to Anthony Bolduc (anthony.bolduc@usherbrooke.ca)
ABSTRACT
In an engineering context, objective evaluation of vibroacoustic models is traditionally performed with visual
or numeric information. However, actual auditory perception cannot be transmitted through these types
of objective representations. Sound Field Reproduction (SFR) of sound ﬁelds emitted by physical objects
is often based on simplistic point source models with modiﬁed radiation properties or from recordings,
using stereophonic or binaural techniques. In the perspective of better perceptual evaluation of engineered
products, it would be useful if such methods would not be limited to certain types of sources, modeling
techniques or predeﬁned listening spots. A general SFR method using Wave Field Synthesis formalism
applied to common vibroacoustic models as found in mechanical engineering is proposed. SFR applied
to an analytical model of a harmonic or broadband excited plate is studied using three Secondary source
distributions geometries. Results of numerical simulations illustrate the viability and limits of the approach.
1. INTRODUCTION
In an engineering context, vibroacoustic models are used
to predict a given device’s acoustical performance. Re-
sults are traditionally evaluated with visual or numeric
information, such as sound spectrums, intensity ﬁelds
or sound pressure levels. This leads to a contradiction
since visual media are employed to make decisions about
auditory problematics. For instance, Transmission Loss
Factor (TLF) of a composite aeronautic panel is a com-
mon engineering measurement that often uses such vi-
sual evaluation media. A larger TLF does not guaran-
tee a better auditory perceptual comfort. To circumvent
this contradiction, perceptual evaluation and sound qual-
ity testing are now an essential part of product develop-
ment, because annoying audible artefacts can be visually
absent fromsuch objective data representation. Extended
Sound Field Reproduction (SFR) of vibroacoustics mod-
els instead of binaural or stereophonic sound presenta-
tion becomes interesting in a jury testing environment.
Many vibroacoustic models rely on advanced computing
methods, like FEMor BEM. These powerful descriptions
of vibroacoustic behaviors are rarely used to drive virtual
auditory displays in a mechanical product design work-
ﬂow. This is the purpose of this paper. The suggested
approach would render virtual reproduction of mock-ups
easier. Perceptual parametric evaluations of vibroacous-
tic models (e.g. an aircraft trim panel) would be possible
even before a prototype would be manufactured [1].
Wave Field Synthesis (WFS) is generally used for SFRof
ﬁelds deﬁned by a sum of plane or spherical waves, with
their origin in a horizontal listening plane. It is known
that WFS can evoke accurate sound localization of the in-
dividual virtual sound sources even when a multi-source
sound ﬁeld is synthesized [2]. To provide a more re-
ﬁned spatial deﬁnition of a virtual source, directivity pat-
terns can be implemented for virtual point sources [3].
Also, signal processing can be applied to give the source
a perceived spatial extent to simulate width, height and
depth [4]. The radiation pattern of a 3D multi-transducer
speaker can be digitally-controlled to approach an ob-
ject’s radiation pattern [5]. Physical modeling of an ob-
ject, like a bowed string or a waveguide, can be employed
to deﬁne a radiation pattern and frequency response and
then be applied to a point source [6].
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Bolduc et al. Sound Field Reproduction of Vibroacoustic Models
Recently, more physically-informed models of plates and
spheres have been imitated by the strategic placement
of virtual point sources at antinodes with proper polar-
ity [7]. However, in [7], these vibrating modes are only
used to provide a distributed source pattern, i.e. they are
not representative of frequency responses of actual plates
and they don’t combine many vibration modes as found
in classical vibroacoustic models. In [7], the selected
modes were carefully chosen in order to circumvent the
problem that arises with vibrating modes that also vary
perpendicular to the horizontal listening plane (e.g. for a
vertical plate). The aim of this paper is to include a com-
plete vibroacoustic model as a virtual source, including:
frequency response, a series of mode shapes and sound
radiation pattern that varies with frequency. SFR of 3D
polyhedron source models and a modiﬁcation to the WFS
operator for out-of-plane virtual sources have been pro-
posed in [8] for musical purposes.
This paper uses an approach to the aforementioned aim
based on the discretization of a vibrating surface of
an object into a high-density grid of monopoles in the
derivation of WFS operators based on Rayleigh I inte-
gral, such as proposed in [8]. This allows a homogeneous
method, independent of the object geometry, which can
be used to reproduce the sound ﬁelds of complex mod-
els solved with analytical or numerical methods. To this
end, a physical approach of sound reproduction was se-
lected as the most appropriate avenue. In this case, the
reproduced sound ﬁeld is only deﬁned through the target
radiated ﬁeld. WFS was selected because of its deﬁni-
tion based on Rayleigh integrals, which are a common
tool for the vibroacoustic engineer. Section 2 presents
the vibroacoustic plate model. Section 3 describes how
WFS is adapted to a plate mode. Simulation results are
provided in Sect. 4.
2. MODEL OF A THIN PLATE
An analytical model of a vertically-oriented rectangu-
lar simply-supported bafﬂed plate was used to obtain the
normal velocity u
A
(x
A
, ω) on the plate surface A [9]. An
aluminum plate (Young’s modulus E
A
= 70 GPa, den-
sity ρ
A
= 2700 kg/m
3
, structural loss factor η = 0.0001
and Poisson’s ratio ν = 0.33) with signiﬁcant dimen-
sions comparatively to the reproduction speaker array
was used: width a = 1.2 m, height b = 0.8 m and thick-
ness e = 3 mm. The plate is mechanically excited by a
harmonic point-force at a frequency f
exc
[Hz]. Normal
velocity ﬁelds u
A
are shown in Fig. 1 for the two fre-
quencies used in the simulations.
Fig. 1: Real part of the normal velocity ﬁeld u
A
of the
plate excited at mode 6-5 (470 Hz, left) and mode 17-5
(1759 Hz, right) by a harmonic point-force in ×.
Half-space conditions are assumed for the deﬁnition
of the target sound ﬁelds, with air as the acous-
tic medium with sound speed c
0
= 343 m/s, density
ρ
0
= 1.2041 kg/m
3
and acoustical wavenumber k =
ω/c
0
[rad/m]. The complex harmonic radiated sound
pressure ﬁeld p(x, ω) for a geometry shown in Fig. 2 is
given by [9]:
p(x, ω) = jωρ
0
__
A
e
−jk||x−x
A
||
2π||x −x
A
||
u
A
dA. (1)
Fig. 2: Coordinate system, plate and Secondary Source
Distribution (SSD) geometry. x is any ﬁeld position, x
0
is
any point on the SSD A
0
, x
A
is any point on the vibrating
surface A, y
0
is the y coordinate of the SSD, y
ref
is the
distance between y
0
and the 2.5D WFS reference line
andn
0
is the A
0
surface normal. The origin is collocated
with the plate’s center at y
A
= 0.
Equation (1) will be used to: 1) derive the WFS SSD
driving function and 2) compute the target sound ﬁeld
in the listening area for comparison with the reproduced
sound ﬁeld. Numerical simulations are carried out by
discretizing the plate’s surface as a regular-spaced grid
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Bolduc et al. Sound Field Reproduction of Vibroacoustic Models
with 61 points along the x axis and 41 points along the
z axis (∆x
A
= ∆z
A
= 2 cm). The modal expansion of u
A
was computed for resonant frequencies up to 4000 Hz.
3. WAVE FIELD SYNTHESIS
WFS is based on the Kirchhoff-Helmholtz integral,
which states that the radiation of sound sources outside
a volume V enclosed by a surface A
0
is uniquely deﬁned
in V by the sound pressure and normal sound pressure
gradient on A
0
. In practical WFS implementation, only
monopole secondary sources are employed as they corre-
spond easily with small closed-box loudspeakers. Thus,
for any SSD, the reproduced sound ﬁeld S is given by
S(x, ω) =−
_
A
0
G(x,x
0
, ω)D(x
0
, ω)dA
0
, (2)
with bafﬂed monopole Green function G =
e
−jk||x−x
0
||
_
(2π||x −x
0
||) and secondary source driv-
ing function D. In the next subsections, different
SSD conﬁgurations that depart from an ideal inﬁnite
continuous SSD are introduced and compared. The
1D Euler equation (Eq. (3) with n
A
=n
0
= [0 1 0] for
this particular problem) and the Kirchhoff-Helmholtz
integral with only the monopole term (Eq. (4)) are used
in the development:
∂ p(x, ω)
_
∂n(x) =−jωρ
0
u(x, ω), (3)
p(x
2
, ω) =−
_
S
1
G(x
2
,x
1
, ω)
∂ p(x
1
, ω)
∂n
1
dA. (4)
The transverse particle velocity ﬁeld u
0
(x
0
, ω) on A
0
is obtained by calculating the pressure ﬁeld at A
0
with
Eq. (4) using ∂ p(x
A
, ω)
_
∂n
A
, then rearranging Eq. (3)
evaluated at A
0
to have Eq. (6):
p(x
0
, ω) =
__
A
G(x
0
,x
A
, ω) jωρ
0
u
A
dA, (5)
u
0
(x
0
, ω) =−( jωρ
0
)
−1
∂ p(x
0
, ω)
_
∂n
0
. (6)
Substituting Eq. (5) in Eq. (6) with r
0
=||x
0
−x
A
|| gives
u
0
=−
__
A
∂G(x
0
,x
A
, ω)
∂n
0
u
A
dA =−
__
A
∂G
∂r
0
∂r
0
∂y
0
u
A
dA
=
__
A
e
−jkr
0
2πr
0
_
1
r
0
+ jk
__
y
0
−y
A
r
0
_
u
A
dA.
(7)
Equation (3) evaluated at A
0
equals D(x
0
, ω) for an in-
ﬁnite continuous planar SSD. In this case, ideal SFR is
achieved in V, i.e. S = p, with Eq. (2) as a surface inte-
gral over A
0
.
3.1. Stationary Phase Approximation and 2.5D
Wave Field Synthesis operator
For an inﬁnite continuous linear SSD targeted at SFR
in a horizontal listening plane, the Stationary Phase Ap-
proximation (SPA) is used [3]. Thus, z
0
= 0 in each x
0
-
dependant function, z = 0 in each x-dependant function
and Eq. (2) becomes a line integral. To keep physical
homogeneity, SPA introduces a factor Q [m] varying for
each degenerated SSD conﬁguration, i.e. linear or curvi-
linear, and depending on the virtual source type. Tradi-
tionally, Qis dependent ofx. Areference line y
ref
parallel
to the SSD, where the amplitude of the reproduced sound
ﬁeld will match the amplitude of the target sound ﬁeld, is
chosen to improve the overall reproduction quality while
making Q independent of x. In this case, the literature
proposes Q =
_
2πy
ref
/ jk. It was shown that this classi-
cal expression for Q is not physically exact when the re-
produced wave is different from a plane wave propagat-
ing in a direction perpendicular to a linear SSD: one must
use Q
PW
= Q
√
sinθ
PW
, depending on the angle θ
PW
be-
tween the linear SSD and the wave front direction, in
order to give results in agreement with the target sound
ﬁeld [10]. Such deviations can reach several dB for wave
fronts that are not parallel to the SSD while crossing the
linear SSD, which is the case of this paper where virtual
point sources with different phases are emitting near and
behind the SSD. As speciﬁed by [10], this type of sys-
tematic error has not been investigated for virtual sound
ﬁelds other than plane waves. In this paper, we rely on
a simple hypothesis: for point sources emitting spherical
waves near and behind the SSD, the factor to use for a
reproduction in a horizontal plane is
Q
SW
=
¸
2πy
ref
jk
_
r
0
y
0
−y
A
+y
ref
, (8)
which is in accordance with [3, Eq. (2.22a)] when 1/r
0
is
not small compared to jk in Eq. (7). This simpliﬁcation
is often made when using SPA for reproduction of plane
waves and virtual point sources far from the SSD, i.e.
1/r
0
→0.
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Bolduc et al. Sound Field Reproduction of Vibroacoustic Models
3.2. Truncation and discretization of the Sec-
ondary Source Distribution
To attenuate corner effects due to the nature of the dis-
crete ﬁnite array, a Tukey window W(x
0
) was applied to
the SSD [11]. Five speakers were weighted on each side,
excluding the zero-weight of both extremities. Compu-
tation of the reproduced sound ﬁeld Eq. (2) becomes
S =−∆x
0
L
0
∑
i=1
W(x
0,i
)G(x,x
0,i
, ω)D(x
0,i
, ω)Q
SW
, (9)
with L
0
secondary sources along the x
0
line.
4. RESULTS OF NUMERICAL SIMULATIONS
Three cases of different SSDs are studied: an inﬁnite
continuous planar array, which is the ideal WFS SSD for
a half-space listening area; an inﬁnite continuous linear
array, which is prescribed by the 2.5DWFS theory for re-
construction in a listening horizontal plane; and a ﬁnite
discrete linear array, which is used for a practical imple-
mentation in a physical setup. On ﬁgures, gray marks (×)
represent discrete secondary sources and a bold gray line
represents a continuous SSD. Those cases show the de-
generation that occurs between the perfectly theoretical
WFS reproduction setup and a practical implementation.
Since numerical computations were used for inﬁnite con-
tinuous arrays, the three cases were discretized as shown
in Tab. 1. Also, y
ref
= 2 m, y
A
= 0 m and y
0
= 1 m. For
the ﬁnite discrete linear array, simulation results are as-
sumed valid between 45 and 1040 Hz, due to the spatial
aliasing theorem [3]:
c
0
2(L
0
−1)∆x
0
≤ f ≤
c
0
2∆x
0
= f
alias
. (10)
Simulations were carried for two cases of plate excita-
tion. One is a unit point force at 470 Hz, belowthe spatial
aliasing frequency ( f
alias
) limit due to the discrete nature
of the SSD. The other is above this limit at 1759 Hz. Ob-
jective evaluation of the reproduced sound ﬁeld by corre-
sponding SSD is given by a normalized error ﬁeld which
accounts for the amplitude and phase errors, deﬁned as
e
r
(x, ω) = 20log
10
_
|S(x, ω) −p(x, ω)|
|p(x, ω)|
_
. (11)
The lower the e
r
, the better the reproduction. The ob-
jective evaluation area range is x = {−2, 2} m and y =
{1, 5} m. Also, an error metric which only accounts for
the amplitude error is deﬁned as
L
0
H
0
∆x
0
A
0
Inf. cont. planar 541 361 2.22 cm 12x8 m
(=∆z
0
)
Inf. cont. linear 1201 1 1 cm 12 m
Fin. discr. linear 24 1 16.5 cm 4 m
Table 1: SSD geometric parameters: L
0
and H
0
are the
numbers of secondary sources along the x
0
line and z
0
line, respectively, ∆x
0
and ∆z
0
are the source separations
and A
0
is the SSD dimension.
Fig. 3: Real part of target sound pressure ﬁelds p in the
x-y plane. Left: Plate excitation at 470 Hz (mode 6-5).
Right: Plate excitation at 1759 Hz (mode 17-5). Points
where frequency responses are computed (Sect. 4.3) are
shown as circles. Plate limits are shown as ♦.
e
a
(x, ω) = 20log
10
_
|S(x, ω)
_
p(x, ω)|
_
. (12)
For e
a
= 0 dB, the reproduction and target amplitudes
match. For e
a
< 0 dB, the reproduction amplitude is
lower than the target amplitude. For e
a
> 0 dB, the re-
production amplitude is higher than the target amplitude.
4.1. Reproduction in the horizontal plane
Target sound ﬁelds in the horizontal x-y plane are shown
in Fig. 3. Reproduced sound ﬁelds are shown for the
three SSD cases in Figs. 4, 5 and 7. The right part of
Fig. 3 reveals noteworthy considerations. It suggests that
a mode-dependent discretization of the plate as proposed
by [7] would work for surface modes, but not edge or
corner modes: the corners of the plate at mode 17-5 con-
tribute more to the overall radiation than its surface.
Reproduced sound ﬁelds in the top parts of Figs. 4 and
7 show that the sound pressure ﬁeld of a 3D plate model
can be reproduced with an ideal SSD, with errors e
r
rang-
ing between -35 and -60 dB in the listening area. Dis-
cretization of the SSD does not have a major contribution
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Bolduc et al. Sound Field Reproduction of Vibroacoustic Models
Fig. 4: Left: real part of reproduced sound pressure ﬁelds
S in the x-y plane. Right: reproduction error e
r
. SSDs
are: Top: inﬁnite continuous planar. Center: inﬁnite
continuous linear, Bottom: ﬁnite discrete linear. Plate
excitation is 470 Hz (mode 6-5).
to the reproduction error: only the 2.5D WFS operator
does (center and bottom of Fig 4). Figure 5 shows that in
reality the exact position of the reference line e
a
= 0 dB
deviates slightly from its theoretical deﬁnition, i.e. a
straight parallel line to the SSD, depending on the sound
ﬁeld, even for an ideal linear SSD.
The bottom part of Fig. 7 shows that spatial aliasing oc-
curs when the excitation frequency is outside the range
deﬁned by Eq. (10). The center part of Fig. 7 is inter-
Fig. 5: Amplitude error e
a
for results shown in Fig. 4.
SSDs are: Left: inﬁnite continuous linear. Right: ﬁnite
discrete linear. The continuous black line is the 0 dB am-
plitude error line. The dashed black line is the theoretical
2.5D WFS reference line.
Fig. 6: Real part of target and reproduced sound ﬁelds in
the x-y plane. Plate excitation is 1759 Hz (mode 17-5).
Gray line: target wave. SSDs are: Black line: inﬁnite
continuous linear. Thin dotted line: ﬁnite discrete linear.
esting: even for an ideal linear SSD, the error is of the
order of the sound ﬁeld (clipped at 0 dB). This is due to
the fact that the mode 17-5 is a corner mode, i.e. part
of the plate’s main radiation arises from outside the x-y
plane. In this case, the trace wavelength in the horizontal
plane is different from λ = 2π/k. This is illustrated in
Fig. 6, which corresponds to a side view of a linear slice
of the bottom part of Fig. 7. Even if phases are aligned at
the SSD location, a π/4 phase difference is noticeable at
the reference line. Also, the phase differences between
the target and reproduced sound ﬁelds are not similar if
one compares discrete and continuous linear SSDs. In
Fig. 6, one can also note the target sound ﬁeld r
−1
-like
and the reproduced sound ﬁeld r
−
1
/
2
-like spatial decays.
At the top and center parts of Fig. 7, even if the repro-
duction error is above 0 dB for the latter, a human ear
may not hear much difference between the sound ﬁelds
shown since the error e
r
is mainly due to the phase er-
ror: one can see that both sound ﬁelds look similar. One
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Bolduc et al. Sound Field Reproduction of Vibroacoustic Models
Fig. 7: Left: reproduced sound pressure ﬁelds S in the
x-y plane. Right: reproduction error e
r
. SSDs are: Top:
inﬁnite continuous planar. Center: inﬁnite continuous
linear. Bottom: ﬁnite discrete linear. Plate excitation is
1759 Hz (mode 17-5).
can note that reproduced sound ﬁelds share many simi-
larities, e.g. left-right presence of the sources at the plate
corners and spatial distribution of the directivity lobes.
4.2. Reproduction in the vertical plane
In order to evaluate what kind of spatial information
might be lost along a vertical plane for a practical hor-
izontal linear WFS setup, target sound pressure ﬁelds in
vertical y-z planes are shown in Fig. 8. For the mode 6-5,
evaluation is given at x = 0.2 m to avoid a null line in
Fig. 8: Real part of target sound pressure ﬁelds p in a
y-z plane. Left: plate excitation at 470 Hz (mode 6-5,
x = 0.2 m). Right: plate excitation at 1759 Hz (mode
17-5, x = 0 m). Plate limits are shown as ♦.
the radiation pattern. For the mode 17-5, x = 0 m. Re-
produced sound ﬁelds are only reported for the inﬁnite
continuous planar and the ﬁnite discrete linear SSDs in
Figs. 9 and 10. The bottom parts of Figs. 9 and 10 show
that a linear SSD cannot faithfully reproduce a modal
sound ﬁeld with z-varying characteristics in the listening
area. The top parts show that such reproductions can be
achieved with 2D SSDs. Below f
alias
, the bottom-right
part of Fig. 9 illustrates why a listener does not need to
be exactly in the reproduction plane to have an accept-
able reproduction experience: there is an extended area
around z = 0 with an error e
r
of the same order.
4.3. Broadband excitation
The previous results illustrated the reproduced sound
ﬁelds in spatial domain for the case of harmonic exci-
tations. In real practical applications, a broadband ex-
citation based on measured or predicted excitation data
would be used to drive the plate model, whether me-
chanical or airborne. This subsection reports the perfor-
mance of the method for a broadband excitation. Sound
pressure was computed for the three locations shown in
Fig. 3 with a ﬁnite discrete linear SSD. Figure 11 shows
the frequency response functions (FRFs) for a ﬁeld point
on the reference line. The target and reproduced am-
plitudes are in good agreement. As expected, the dif-
ference is stronger between resonant frequencies or for
anti-resonances. However, since the latter also corre-
sponds to softer sounds, it is expected that these differ-
ences should not be perceived, i.e. they would tend to be
masked by stronger resonating frequencies. Above f
alias
,
the reproduced FRF approaches the target FRF. Although
less precisely than below f
alias
, the correspondence is still
acceptable. Therefore, we expect that the plate’s timbre
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Bolduc et al. Sound Field Reproduction of Vibroacoustic Models
Fig. 9: Left: real part of reproduced sound pressure ﬁelds
S in a y-z plane at x = 0.2 m. Right: reproduction error
e
r
(clipped positive values range from 0 to 18 dB). SSDs
are: Top: inﬁnite continuous planar. Bottom: ﬁnite dis-
crete linear. Plate excitation is 470 Hz (mode 6-5).
should be appropriately reproduced even if WFS aliasing
artefacts exist.
2.5D WFS theory predicts that the amplitude of the re-
produced sound ﬁeld is, on the average, greater than the
target amplitude between the SSD and the reference line,
the same on the reference line, and lower after the refer-
ence line. Figure 12 illustrates this behavior for the re-
ported test case. Also, it shows that the amplitude error
is more acceptable and less distributed between approxi-
mately 200 and 600 Hz, instead of the aliasing frequency
limits deﬁned by Eq. (10), i.e. 45 and 1040 Hz. This is
believed to be the result of the plate model geometry: for
a wider-than-high rectangular horizontal plate, more hor-
izontal modes will contribute in the low spectrum, while
more and more vertical modes will contribute to the over-
all radiation with an increasing excitation frequency.
5. CONCLUSION
This paper presents a general method for sound ﬁeld
Fig. 10: Left: real part of reproduced sound pressure
ﬁelds S in the y-z plane. Right: reproduction error e
r
(clipped positive values range from 0 to 20 dB). SSDs
are: Top: inﬁnite continuous planar. Bottom: ﬁnite dis-
crete linear. Plate excitation is 1759 Hz (mode 17-5).
reproduction of vibroacoustic models with Wave Field
Synthesis. The harmonic sound pressure ﬁeld radiated
by an analytical model of a plate is reproduced with three
geometries of Secondary Source Distributions for two
excitation frequencies, i.e. under and over the spatial
aliasing frequency limit of a practical WFS setup. Com-
parisons are made both in horizontal and vertical listen-
ing planes. Simulations have shown that under the spa-
tial aliasing frequency, the proposed method appropri-
ately reproduces complex radiation patterns in the hor-
izontal listening plane. Broadband simulations suggest
that the timbre of the vibroacoustic model should be ap-
propriately reproduced even if WFS aliasing artefacts ex-
ist. Using the WFS capability to accurately position vir-
tual sources in space, exterior and focused plate mod-
els of signiﬁcant dimensions compared to the speaker ar-
ray will be reproduced by an enclosing 96 WFS system
for objective and perceptual evaluations, mainly for en-
gineering purposes, as opposed to research done in [8].
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Bolduc et al. Sound Field Reproduction of Vibroacoustic Models
Fig. 11: Magnitude of frequency response functions from excitation input force [N] to target (bold gray line) and
reproduced (thin black line) sound [dB] at x
2
(−1, 3, 0), i.e. on the reference line, for a ﬁnite discrete linear SSD.
Fig. 12: Amplitude error e
a
of frequency response func-
tion (black line) at three listening points in the x-y plane
for a ﬁnite discrete linear SSD: Top: x
1
(−0.4, 2, 0).
Center: x
2
(−1, 3, 0) (point on reference line). Bottom:
x
3
(−0.8, 4, 0). The point locations are shown in Fig. 3.
Bold gray line: average value over 200-600 Hz.
6. ACKNOWLEDGMENTS
This work was supported by grants from the Natural Sci-
ences and Engineering Research Council of Canada and
from the University of Sherbrooke.
7. REFERENCES
[1] P.-A. Gauthier, C. Camier, F.-A. Lebel, Y. Pasco
and A. Berry, “Experiments of sound ﬁeld repro-
duction inside aircraft cabin mockup,” presented at
the AES 133rd Convention, San Francisco, United
States, 2012.
[2] E. Start, “Direct sound enhancement by Wave Field
Synthesis,” Ph.D. thesis, Delft University of Tech-
nology, Delft, The Netherlands, 1997.
[3] E. Verheijen, “Sound reproduction by Wave Field
Synthesis,” Ph.D. thesis, 2nd Edition, Delft Univer-
sity of Technology, Delft, The Netherlands, 2010.
[4] M. Laitinen, T. Pihlajamki, C. Erkut, and V. Pulkki,
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[5] O. Warusfel and N. Misdariis, “Sound Source Ra-
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[6] P. Cook, Real sound synthesis for interactive appli-
cations, A K Peters, Natick, 2002, 263 p.
[7] J. Ahrens and S. Spors, “Two Physical Models for
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United States, 2011.
[8] M. Baalman, “On Wave Field Synthesis and
electro-acoustic music, with a particular focus
on the reproduction of arbitrarily shaped sound
sources,” Ph.D. thesis, Berlin University of Tech-
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[9] M. Junger and D. Feit, Sound, Structures, and Their
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[10] J. Ahrens, Analytic methods of sound ﬁeld synthe-
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